Method and apparatus for modeling the modal properties of optical waveguides

ABSTRACT

A method and apparatus models one or more electromagnetic field modes of a waveguide. The method includes sampling a two-dimensional cross-section of the waveguide. The method further includes calculating a first matrix having a plurality of elements and having a first bandwidth using the sampled two-dimensional cross-section of the waveguide. The plurality of elements of the first matrix represents an action of Maxwell&#39;s equations on a transverse magnetic field within the waveguide. The method further includes rearranging the plurality of elements of the first matrix to form a second matrix having a second bandwidth smaller than the first bandwidth. The method further includes shifting the second matrix and inverting the shifted second matrix to form a third matrix. The method further includes calculating one or more eigenvalues or eigenvectors of the third matrix corresponding to one or more modes of the waveguide.

RELATED APPLICATIONS

The present application claims priority to U.S. Provisional ApplicationNo. 60/608,765, filed Sep. 11, 2004, which is incorporated in itsentirety by reference herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to methods for modeling theproperties of waveguides, and more particularly, to methods for modelingthe optical properties of optical waveguides, including but not limitedto optical fibers.

2. Description of the Related Art

In a photonic-bandgap fiber (PBF), the cladding is a photonic crystalmade of a two-dimensional periodic refractive index structure thatcreates bandgaps, namely regions in the optical frequency or wavelengthdomain where propagation is forbidden. This structure can be either aseries of concentric rings of alternating high and low indices, as inso-called Bragg fibers (see, e.g., Y. Xu, R. K. Lee, and A. Yariv,Asymptotic analysis of Braggfibers, Optics Letters, Vol. 25, No. 24,Dec. 15, 2000, pages 1756-1758) or a two-dimensional periodic lattice ofair holes arranged on a geometric pattern (e.g., triangular), as inair-core PBFs. Examples of air-core PBFs are described by variousreferences, including but not limited to, P. Kaiser and H. W. Astle,Low-loss single material fibers made from pure fused silica, The BellSystem Technical Journal, Vol. 53, No. 6, July-August 1974, pages1021-1039; and J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev,Photonic crystal fibers: A new class of optical waveguides, OpticalFiber Technology, Vol. 5, 1999, pages 305-330.

The air hole that forms the core of an air-core PBF is typically largerthan the other holes in the PBF and constitutes a defect in the periodicstructure of the PBF which introduces guided modes within the previouslyforbidden bandgap. PBFs offer a number of unique optical properties,including nonlinearities about two orders of magnitude lower than in aconventional fiber, a greater control over the mode properties, andinteresting dispersion characteristics, such as modal phase velocitiesgreater than the speed of light in air. For these reasons, in opticalcommunications and sensor applications alike, PBFs offer excitingpossibilities previously not available from conventional (solid-core)fibers.

Since the propagation of light in air-core fibers is based on a verydifferent principle than in a conventional fiber, it is important todevelop a sound understanding of the still poorly understood propertiesof air-core fibers, to improve these properties, and to tailor them tospecific applications. Since the fairly recent invention of air-corefibers (see, e.g., J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J.Russell, and J. P. Sandro, “Photonic crystals as optical fibers-physicsand applications,” Optical Materials, 1999, Vol. 11, pages 143-151),modeling has been an important tool for analyzing the modal propertiesof air-core PBFs, particularly their phase and group velocity dispersion(see, e.g., B. Kuhlmey, G. Renversez, and D. Maystre, Chromaticdispersion and losses of microstructured optical fibers, Applied Optics,Vol. 42, No. 4, 1 Feb. 2003, pages 634-639; K. Saitoh, M. Koshiba, T.Hasegawa, and E. Sasaoka, Chromatic dispersion control in photoniccrystal fibers: application to ultra-flattened dispersion, OpticsExpress, Vol. 11, No. 8, 21 Apr. 2003, pages 843-852; and G. Renversez,B. Kuhlmey, and R. McPhedran, Dispersion management with microstructuredoptical fibers: ultraflattened chromatic dispersion with low losses,Optics Letters, Vol. 28, No. 12, Jun. 15, 2003, pages 989-991),transmission spectra (see, e.g., J. C. Knight, T. A. Birks, P. St. J.Russell, and D. M. Atkin, All-silica single mode optical fiber withphotonic crystal cladding, Optics Letters, Vol. 21, No. 19, Oct. 1,1996, pages 1547-1549; and R. S. Windeler, J. L. Wagener, and D. J.Giovanni, Silica-air microstructured fibers: Properties andapplications, Optical Fiber Communications Conference, San Diego, 1999,pages FG1-1 and FG1-2), unique surface modes (see, e.g., D. C. Allan, N.F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman,J. A. West, P. Zhang, and K. W. Koch, Surface modes and loss in air-corephotonic band-gap fibers, Proceedings of SPIE Vol. 5000, 2003, pages161-174; K. Saitoh, N. A. Mortensen, and M. Koshiba, Air-core photonicband-gap fibers: the impact of surface modes, Optics Express, Vol. 12,No. 3, 9 Feb. 2004, pages 394-400; J. A. West, C. M. Smith, N. F.Borelli, D. C. Allan, and K. W. Koch, Surface modes in air-core photonicband-gap fibers, Optics Express, Vol. 12, No. 8, 19 Apr. 2004, pages1485-1496; M. J. F. Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S.Kino, Simple geometric criterion to predict the existence of surfacemodes in air-core photonic-bandgap fibers, Optics Express, Vol. 12, No.9, 3 May 2004, pages 1864-1872; and H. K. Kim, J. Shin, S. H. Fan, M. J.F. Digonnet, and G. S. Kino, Designing air-core photonic-bandgap fibersfree of surface modes, IEEE Journal of Quantum Electronics, Vol. 40, No.5, May 2004, pages 551-556), and propagation loss (see, e.g., T. P.White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel,Confinement losses in microstructured optical fibers, Optics Letters,Vol. 26, No. 21, Nov. 1, 2001, pages 1660-1662; D. Ferrarini, L.Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, Leakage properties ofphotonic crystal fibers, Optics Express, Vol. 10, No. 23, 18 Nov. 2002,pages 1314-1319; and K. Saitoh and M. Koshiba, Leakage loss and groupvelocity dispersion in air-core photonic bandgap fibers, Optics Express,Vol. 11, No. 23, 17 Nov. 2003, pages 3100-3109). However, extensive andsystematic parametric studies of these fibers are lacking becausesimulations of these complex structures are conducted with analyticalmethods that are either time-consuming, or are not applicable to all PBFgeometries. For example, simulating all the modes of a typicallysingle-mode or a few-moded air-core PBF using code currently in use, forexample, the MIT Photonic-Band (MPB) code for computing dispersionrelations and electromagnetic profiles of the modes of periodicdielectric structures, takes approximately 10 hours on a supercomputerusing 16 parallel 2-GHz processors, with a cell size of 10Λ×10Λ and aspatial resolution of Λ/16, where Λ is the period of thephotonic-crystal cladding. (See, e.g., S. G. Johnson, and J. D.Joannopoulos, Block-iterative frequency-domain methods for Maxwell'sequations in planewave basis, Optics Express, Vol. 8, No. 3, 29 Jan.2001, pages 173-190 for further information regarding the MPB code.)Faster computations have been reported using a polar-coordinatedecomposition method to model the fundamental mode (effective index andfields) in six minutes (see, e.g., L. Poladian, N. A. Issa, and T. M.Monro, “Fourier decomposition algorithm for leaky modes of fibres witharbitrary geometry,” Optics Express, 2002, Vol. 10, pages 449-454).However, this high-speed was accomplished by limiting the number of airhole layers to one, and it still required a supercomputer.

In addition, a multipole decomposition method (see, e.g., T. P. White,R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke,“Calculations of air-guided modes in photonic crystal fibers using themultipole method,” Optics Express, 2001, Vol. 8, pages 721-732) has beenable to model all the modes in a hexagonal PBF with four rings of holesat one wavelength in approximately one hour on a 733-MHz personalcomputer. This figure would be reduced to a few minutes on a fasterpersonal computer (e.g., approximately 15 minutes on a 3.2-GHzprocessor.) While this multipole method yields accurate and quickcalculations of the modes of PBFs with circular or elliptical holeshapes, it is limited to these shapes and does not work for PBFs withother hole shapes, for example, for PBFs with scalloped cores (which canadvantageously avoid undesirable surface modes) or nearly-hexagonalcladding holes (a shape which often occurs in actual air-core fibers).In addition, much finer resolutions are needed to accurately model theextremely small physical features of air-core fibers, in particulartheir very thin membranes (e.g., often thinner than Λ/100).

PBFs are typically modeled using numerical methods originally developedfor the study of photonic crystals. Most methods are based on solvingthe photonic-crystal master equation:

$\begin{matrix}{{\nabla{\times \left\{ {\frac{1}{ɛ(r)}{\nabla{\times {H(r)}}}} \right\}}} = {\frac{\omega^{2}}{c^{2}}{H(r)}}} & (1)\end{matrix}$where r is a vector (x, y) that represents the coordinates of aparticular point in a plane perpendicular to the fiber axis, ∈(r)=n²(r)is the dielectric permeability of the fiber cross-section at this point,where n(r) is the two-dimensional refractive index profile of the fiber,H(r) is the mode's magnetic field vector, ω is the optical frequency;and c is the speed of light in vacuum. This formulation is most usefulfor three-dimensional (3D) photonic-bandgap structures, and it is solvedby assuming a constant value for the propagation constant k_(z) andcomputing the eigenfrequencies ω for this value.

Various methods have previously been used to solve for modes in thecontext of air-core PBFs. One previously-used method utilizes the MITPhotonic Bandgap (MPB) code or software which is cited above. Using theMPB code, the solutions of Equation 1 are obtained in the spatialFourier domain, so the first step is to decompose the mode fields and1/∈(r) in spatial harmonics. Equation 1 is then written in a matrix formand solved by finding the eigenvalues of this matrix. This calculationis done by setting the wavenumber k_(z) to some value (e.g., k₀) and bysolving for the frequencies at which a mode with this wavenumber occurs.

Another previously-used method is the beam-propagation method (BPM)(see, e.g., M. Qiu, “Analysis of guided modes in photonic crystal fibersusing the finite-difference time-domain method,” Microwave OpticsTechnology Letters, 2001, Vol. 30, pages 327-330). In the BPM, apseudo-random field is launched into the fiber, which excites differentmodes with different effective indices. The propagation axis istransformed into an imaginary axis, so that the effective indices becomepurely imaginary and the fiber modes become purely lossy. Instead ofaccumulating phase at different rates, the modes now attenuate atdifferent rates, which allow the BPM method to separate them and extractthem iteratively by propagating them through a length of fiber. The modewith the lowest effective index, which attenuates at the smallest rate,is extracted first, and the other modes are computed in order ofincreasing effective index.

Another previously-used method is the finite domain time differencemethod (FDTD), which solves the time-dependent Maxwell's equations:

$\begin{matrix}\left\{ \begin{matrix}{{\nabla{\times E}} = {{- \mu_{0}}\frac{\partial H}{\partial t}}} \\{{\nabla{\times H}} = {ɛ_{0}{ɛ(r)}\frac{\partial E}{\partial t}}}\end{matrix} \right. & (2)\end{matrix}$Solving these equations requires considering the time variable as well(see, e.g., J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C.Knight, P. J. Roberts, and P. St. J. Russell, Robust photonic band gapsfor hollow core guidance in PBF made from high index glass, OpticsExpress, Vol. 11, No. 22, 3 Nov. 2003, pages 2854-2861).

Another previously-used method involves using a two-dimensional modeeigenvalue equation, where the eigenvalue is the mode effective index.The mode is then expanded either in plane-waves (see, e.g., K. Saitohand M. Koshiba, “Full-vectorial imaginary-distance beam propagationmethod based on finite element scheme: Application to photonic crystalfibers,” IEEE Journal of Quantum Electronics, 2002, Vol. 38, pages927-933) or in wavelet functions (see, e.g., W. Zhi, R. Guobin, L.Shuqin, and J. Shuisheng, Supercell lattice method for photonic crystalfibers, Optics Express, Vol. 11, No. 9, 5 May 2003, pages 980-991). Therefractive index profile is decomposed in Fourier components, and theresulting matrix equation is solved for eigenvalues and eigenvectors.

When applied to the calculation of the modes of a PBF, the foregoingpreviously-used methods suffer from a number of disadvantages. Onedisadvantage is that computation times can be extremely long. Anotherdisadvantage is that techniques that rely on a mode expansion requirecalculating the Fourier transform of the refractive index profile of thefiber. For PBFs and other types of waveguides, the refractive indexprofile contains very fine features (e.g., thin membranes) and abruptrefractive index discontinuities (e.g., at the air-glass interfaces),both of which are key to the mode behavior, so a large number ofharmonics should ideally be retained to faithfully describe thesefeatures. However, because retaining these large numbers of harmonicswould require much more memory and computer processing power than isavailable, even in supercomputers, the discontinuities in the refractiveindex profile are smoothened, which artificially widens the finefeatures (e.g., membranes). This standard practice results insignificant differences between the refractive index profile that isbeing modeled and the actual refractive index profile, which in turnintroduces systematic errors in the calculations.

Another disadvantage is that some existing mode solvers must calculatethe modes of the highest order bands first and work their way down tolower-order modes, and this process propagates computation errors. Thisrequirement also increases computation time, since it forces thecomputer to calculate more modes than the ones of interest. In addition,as described above, the multipole decomposition method only works for afew specific hole shapes (e.g., circular and elliptical), thus makingthis technique less versatile.

SUMMARY OF THE INVENTION

In certain embodiments, a method models one or more electromagneticfield modes of a waveguide. The method comprises sampling atwo-dimensional cross-section of the waveguide. The method furthercomprises calculating a first matrix comprising a plurality of elementsand having a first bandwidth using the sampled two-dimensionalcross-section of the waveguide. The plurality of elements of the firstmatrix represents an action of Maxwell's equations on a transversemagnetic field within the waveguide. The method further comprisesrearranging the plurality of elements of the first matrix to form asecond matrix having a second bandwidth smaller than the firstbandwidth. The method further comprises shifting the second matrix andinverting the shifted second matrix to form a third matrix. The methodfurther comprises calculating one or more eigenvalues or eigenvectors ofthe third matrix which correspond to one or more modes of the waveguide.

In certain embodiments, a computer system models one or moreelectromagnetic field modes of a waveguide. The computer systemcomprises means for sampling a two-dimensional cross-section of thewaveguide. The computer system further comprises means for calculating afirst matrix using the sampled two-dimensional cross-section of thewaveguide. The first matrix comprises a plurality of elements and has afirst bandwidth. The plurality of element of the first matrix representsan action of Maxwell's equations on a transverse magnetic field withinthe waveguide. The computer system further comprises means forrearranging the plurality of elements of the first matrix to form asecond matrix having a second bandwidth smaller than the firstbandwidth. The computer system further comprises means for shifting thesecond matrix and inverting the shifted second matrix to form a thirdmatrix. The computer system further comprises means for calculating oneor more eigenvalues or eigenvectors of the third matrix which correspondto one or more modes of the waveguide.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram of an exemplary method compatible with certainembodiments described herein.

FIG. 2 illustrates a first matrix M, which represents the mode equationfor a PBF with a rectangular boundary.

FIG. 3 illustrates a second matrix N obtained by rearrangement of thefirst matrix M to reduce matrix bandwidth.

FIG. 4 is a flow diagram illustrating an exemplary method for modeling aPBF in accordance with certain embodiments described herein.

FIG. 5 illustrates sampling of a refractive index structure used formodeling an air-core PBF having a cladding air hole radius ρ=0.47Λ and acore radius R=0.8Λ under periodic boundary conditions.

FIG. 6 illustrates a calculated band profile and dispersion of the twofundamental modes for the PBF illustrated by FIG. 5 having ρ=0.47Λ andR=1.0Λ.

FIG. 7 illustrates a calculated mode intensity profile of a core modefor the PBF illustrated by FIG. 5 at ρ=0.47Λ, R=0.8Λ, and λ=0.60Λ.

FIG. 8 illustrates a cross section of the core mode of FIG. 7 in a linealong x=0.

FIG. 9 illustrates the convergence for the exemplary method forincreasing grid sizes for a PBF having ρ=0.47Λ and R=0.8Λ(diamond-shaped data points, referring to the leftmost vertical axis)and the convergence for the exemplary method for increasing grid sizesfor the fiber described by Zhu et al. (square-shaped data points,referring to the rightmost vertical axis).

FIG. 10 shows a calculated band profile and dispersion for a PBF havingρ=0.47Λ and R=1.15Λ.

FIG. 11 illustrates a calculated mode profile of a surface mode for thePBF of FIG. 10 at ρ=0.47Λ, R=1.15Λ and λ=0.60Λ.

DETAILED DESCRIPTION

Certain embodiments described herein advantageously provide the abilityto quickly and accurately model PBFs of any geometry, as well as otherwaveguides with complex structures and fine features. For example,certain embodiments described herein are compatible for modeling PBFshaving a single core or multiple cores, and comprising air holes, silicamaterial, or other materials. In certain embodiments, finite-differenceor finite-element calculations are advantageously used in combinationwith a shift-invert technique to solve a magnetic field eigenvalueequation in a sparse matrix form to find the modes (e.g., core, surface,ring, and bulk modes) of a PBF having an arbitrary index profile. Incertain embodiments, a technique for advantageously reducing thebandwidth of the sparse matrix is applied, which advantageously furtherreduces the computation time (e.g., by a factor of 10). As compared to aplane-wave expansion, certain embodiments described hereinadvantageously reduce the computation time by about three orders ofmagnitude and advantageously cut down the amount of required datastorage by about two orders of magnitude. Certain such embodimentsadvantageously enable full simulations of all the mode properties of agiven fiber to be calculated quickly on a personal computer. Inaddition, certain embodiments described herein are simple to implement(e.g., using only approximately 200 lines of code in MATLAB®, availablefrom The MathWorks of Natick, Mass.).

In finite-difference or finite-element calculations, the waveguide indexprofile is sampled (or digitized) and the gradients involved in the modeequation are approximated by finite-difference equations. Suchfinite-difference or finite-element calculations have previously beensuccessfully applied to holey fibers and other waveguides (see, e.g., Z.Zhu and T. G. Brown, “Full-vectorial finite-difference analysis ofmicrostructured optical fibers,” Optics Express, 2002, Vol. 10, pages853-864; and C. Yu and H. Chang, “Applications of the finite differencemode solution method to photonic crystal structures,” Optics And QuantumElectronics, 2004, Vol. 36, pages 145-163), but they have not yet beenadapted for modeling PBFs.

Certain embodiments described herein advantageously model the fields andpropagation constants of the modes (e.g., core, cladding, surface modes,etc.) of a waveguide with an arbitrary index profile n(x,y) that istranslation invariant along the waveguide's longitudinal axis (z). Forexample as described more fully below, in certain embodiments, thewaveguide comprises an optical fiber. As is well known in the art, thetransverse magnetic field h_(T)(r) of the waveguide modes satisfies theeigenvalue equation:{∇_(T) +k ₀ ²(n ²(x,y)−n _(eff) ²)}h _(T)=(∇_(T) ×h _(T))×∇_(T)(1n n²(x,y))  (3)where

$k_{0} = \frac{2\pi}{\lambda_{0}}$(λ₀ is the wavelength), h_(T)=h_(x)(x,y)u_(x)+h_(y)(x,y)u_(y) is thetransverse magnetic field, h_(x) and h_(y) are the components of thetransverse magnetic field projected onto the unit vectors u_(x) andu_(y),

${\nabla_{T}{= {{\frac{\partial}{\partial x}u_{x}} + {\frac{\partial}{\partial y}u_{y}}}}};$and n_(eff) is the mode effective index. With this notation, the totalmagnetic field vector H is simply h_(T)+h_(z)u_(z), where u_(z) is theunit vector along the fiber z axis and h_(z) is the longitudinalmagnetic field. Equation 3 is satisfied by all modes (TE, TM, and hybridmodes). The longitudinal component of the magnetic field and thecomponents of the electric field can all be calculated from thetransverse magnetic field using:

$\begin{matrix}\left\{ \begin{matrix}{h_{z} = {\frac{i}{k_{0}n_{eff}}{\nabla_{T}{\cdot h_{T}}}}} \\{e_{T} = {{- \left( \frac{\mu_{0}}{ɛ_{0}} \right)^{1/2}}\frac{1}{k_{0}n^{2}}u_{z} \times \left\{ {{k_{0}n_{eff}h_{T}} - {\frac{1}{k_{0}n_{eff}}{\nabla_{T}\left( {\nabla_{T}{\cdot h_{T}}} \right)}}} \right\}}} \\{e_{z} = {{i\left( \frac{\mu_{0}}{ɛ_{0}} \right)}^{1/2}\frac{1}{k_{2}n^{2}}{u_{z} \cdot \left( {\nabla_{T}{\times h_{T}}} \right)}}}\end{matrix} \right. & (4)\end{matrix}$

Equation 3 is vectorial and has two equations that couple the twocomponents h_(x)(x,y) and h_(y)(x,y) of the transverse magnetic field,and thus they are solved simultaneously. Note that the form of Equation3 is such that it can be solved by fixing the wavelength and calculatingall the fiber modes at this wavelength.

FIG. 1 is a flow diagram of an exemplary method 100 for modeling one ormore electromagnetic field modes of a waveguide in accordance withcertain embodiments described herein. In an operational block 110, atwo-dimensional cross-section of the waveguide is sampled. In anoperational block 120, a first matrix comprising a plurality of elementsand having a first bandwidth is calculated using the sampledtwo-dimensional cross-section of the waveguide. The plurality ofelements of the first matrix represents an action of Maxwell's equationson a transverse magnetic field within the waveguide. In an operationalblock 130, the plurality of elements of the first matrix is rearrangedto form a second matrix having a second bandwidth smaller than the firstbandwidth. In an operational block 140, the second matrix is shifted andinverted to form a third matrix. In an operational block 150, one ormore eigenvalues or eigenvectors of the third matrix are calculated. Theone or more eigenvalues or eigenvectors correspond to the one or moremodes of the waveguide.

In certain embodiments, sampling the two-dimensional cross-section ofthe waveguide in the operational block 110 comprises digitizing arefractive index profile of the waveguide. For example, for an air-corePBF, sampling the two-dimensional cross-section of the fiber comprisesdigitizing the features of the fiber (e.g., the core, the claddingstructure comprising the air holes and the intervening membranes) in aplanar cross-section which is perpendicular to the fiber longitudinalaxis. In certain embodiments, sampling is done over an area thatreflects the crystal periodicity (e.g., over an area corresponding to aminimum cell that is a fundamental component of the cladding'sphotonic-crystal structure). In certain embodiments, sampling includesdefining the boundary conditions (e.g., periodic boundary conditions,fixed boundary conditions). Depending on (i) the level of complexity ofthe fiber profile, (ii) the desired modeling accuracy, and (iii) thescanning accuracy, this sampling can be carried out in certainembodiments by using a scanner and digitizing the scanned image using acomputer, or in certain other embodiments by entering the fiber profileas a series of mathematical objects (e.g., a pattern of circles of givenradii centered on a periodic grid in the case of a particular air-corefiber) and digitizing this mathematical construct with a computer.

In certain embodiments, calculating the first matrix M in theoperational block 120 comprises discretizing the eigenvalue equation(e.g., Equation 3). This discretization is performed in certainembodiments using the following expressions of Maxwell's equations:

$\begin{matrix}\left\{ \begin{matrix}{{\overset{->}{\nabla}{\times \overset{->}{E}}} = \frac{- {\partial\overset{->}{B}}}{\partial t}} \\{{\overset{->}{\nabla}{\times \overset{->}{H}}} = \frac{\partial\overset{->}{D}}{\partial t}} \\{{\nabla{\cdot \overset{->}{H}}} = 0}\end{matrix} \right. & (5)\end{matrix}$These expressions of Maxwell's equations advantageously yield a simplerexpression for the discretized Maxwell equation matrix, as well as afaster computation. Other expressions of Maxwell's equations may be usedin other embodiments.

In certain such embodiments, the discretization of the eigenvalueequation (e.g., Equation 3) is performed using a finite-differencetechnique based on Yee's cell to account for the field and refractiveindex discontinuities (see, e.g., Z. Zhu and T. G. Brown,“Full-vectorial finite-difference analysis of microstructured opticalfibers,” Optics Express, 2002, Vol. 10, pages 853-864). In certainembodiments, the discretization further comprises index-averaging overeach discretization pixel that straddles an air-core boundary. Incertain embodiments, index-averaging comprises assigning a refractiveindex to a pixel which is equal to a weighted average of the refractiveindices of the media that are straddled by the pixel, in proportion tothe relative surface of the media. For example, if the modeled waveguideis a photonic-bandgap fiber comprising air holes in silica (two mediaonly), the index assigned by index-averaging to a pixel that straddles20% of air and 80% of silica is n_(aver)=0.2n_(air)+0.8n_(silica), wheren_(ave) is the index-averaged refractive index, n_(air) is therefractive index of air, and n_(silica) is the refractive index ofsilica. In certain other embodiments, index-averaging comprises using aharmonic average by calculating the inverse of the average of theinverses of the indices. Other index-averaging techniques are alsocompatible with embodiments described herein. In certain suchembodiments, index-averaging advantageously improves the accuracy of thecomputation at the cost of smoothing the refractive index structure overa small percentage of the total number of pixels.

In certain embodiments, after discretization, the eigenvalue equationcan be written as:Π h_(T)=k₀ ²n_(eff) ²h_(T)  (6)In Equation 6, Π is a linear operator acting on a vector field. Incertain embodiments, to solve this equation, the eigenvalue problem isdiscretized by sampling the two fields h_(x)(x, y) and h_(y)(x,y) in aplanar cross-section perpendicular to the fiber longitudinal axis on agrid of m_(x) points in the x direction and my points in the ydirection. This operation yields a new magnetic field vector h(x, y) ofdimension 1×2m_(x)m_(y), which contains (arranged using some arbitrarynumbering system) S=m_(x)m_(y) samples of h_(x)(x, y) and S samples ofh_(y)(x, y). In certain embodiments, this sampling is done over an areathat reflects the crystal periodicity (e.g., over an area correspondingto a minimum cell that is a fundamental component of the fibercross-section's photonic-crystal structure). In certain embodiments,this sampling is performed over the entire fiber cross-section.

In certain embodiments, the first matrix is calculated in theoperational block 120 by discretizing the linear operator Π usingfinite-difference equations. In certain embodiments, discretizing thelinear operator Π entails sampling the refractive index profile n(x, y)of the fiber, so that it is represented by a first matrix M of dimension2S×2S. In certain embodiments, this sampling is carried out on the sameS×S grid as the grid used to sample the magnetic field.

In certain embodiments, the eigenvalue equation, as expressed byEquation 6, is solved subject to standard boundary conditions. Unlikewith a k-domain method, which relies on the Bloch theorem and thus mustuse periodic boundary conditions, certain such embodiments can use oneof the following types of boundary conditions: (1) fields vanishingoutside the simulation area; (2) periodic boundary conditions, namelyfiber cross-section and fields periodically repeated over the entirespace, with the simulation domain as a unit supercell; (3) surroundingthe simulation domain by an absorbing material (e.g., for propagationloss calculation); or (4) perfectly matching boundary conditions. Otherembodiments can utilize other boundary conditions, depending on thewaveguide being modeled. In certain embodiments, both the boundaryconditions and the simulation domain boundary are taken into accountwhen sampling the operator Π, so that this information is also containedin the first matrix M itself.

In certain embodiments, the eigenvalue equation can be expressed as asingle matrix equation:M h=k₀ ²n_(eff) ²h  (7)As described above, the first matrix M is calcuated from the linearoperator Π by discretizing the eigenvalue equation. Finding the roots ofEquation 7 (e.g., the eigenvalues n_(eff) and the eigenmodes or fielddistributions of the fiber modes) is a typical matrix eigenvalueproblem. However, the operator matrix M is typically extremely large.For example, consider the case of a photonic-bandgap fiber with atriangular pattern of circular air holes of radius ρ=0.47Λ, and acircular core of radius R=0.8Λ, 8 rows of air holes in the cladding, anda square cladding boundary. The cell is then sixteen periods on eachside, therefore it has dimensions of 16Λ×16Λ. To resolve the fiber'sthin membranes and achieve sufficient accuracy, a grid step size (orresolution) of Λ/30 can be advantageously used. The number of samplingpoints is then m_(x)=16×30 in x and m_(y)=16×30 in y. ThusS=m_(x)m_(y)=230,400, and the first matrix M contains (2S)²≈212 billionelements. Finding the eigenvalues of matrices this large without usingmethods compatible with certain embodiments described herein is out ofthe range of most computers.

However, the replacement of the differential operators by theirfinite-difference equivalents makes the first matrix M a sparse matrix.FIG. 2 illustrates the general form of the first matrix M representingthe mode equation for the above example. The horizontal and verticalaxes of FIG. 2 represent the coordinate of the elements in the firstmatrix M, which each go up to a maximum value of 2S=460,800. The soliddots, which merge into solid curves where their density is too high tobe individually resolved in FIG. 2, represent non-zero elements. Theblanks between these elements are all zero elements. In this particularexample, the total number of non-zero elements n_(z) is 2,370,736. Thematrix density D of this type of matrix, defined as the ratio ofnon-zero elements to the total number of elements, is

${\frac{5}{2S} \leq D \leq \frac{7}{2S}},$the exact value depending on the index profile. In the foregoingexample, the matrix density is only of the order of D≈1.1×10⁻⁵, orequivalently, only about 0.0011% of the matrix elements are non-zero.The implication is that in spite of the large size of the first matrixM, the number of elements that must be stored during computation iscomparatively low, down from approximately 212 billion to approximately2.4 million for this typical example, which is well within the range ofwhat can be handled with standard personal computers.

Another parameter of the first matrix M is its bandwidth B_(M), which isdefined as the maximum distance between a non-zero coefficient and thefirst diagonal:

$\begin{matrix}{B_{M} = {\max\limits_{{({i,j})};{{M{({i,j})}} \neq 0}}{{i - j}}}} & (8)\end{matrix}$Small bandwidths are advantageous for the fast computation ofeigenvalues in sparse matrices. The bandwidth of the first matrix M isproportional to S and can be quite large, as illustrated in FIG. 2. Incertain embodiments, the bandwidth is advantageously lowered in theoperational block 130 by rearranging the (2S)² elements of the firstmatrix M to form a second matrix N having a bandwidth smaller than thatof the first matrix M. In certain such embodiments, rearranging thefirst matrix M comprises applying a permutation on the elements of bothh and M (see, e.g., A. George and J. Liu, “Computer Solution of LargeSparse Positive Definite Systems,” Prentice-Hall, 1981). In certain suchembodiments, this operation leads to a second matrix N in which all thenon-zero elements are clustered near the first diagonal and thebandwidth is therefore greatly reduced. The second sparse matrix N andthe new permutated magnetic field vector g produced by this operationalso satisfy Equation 7 (e.g., N g =K₀ ²n_(eff) ²g). In certainembodiments, the second matrix N has the same dimensions as the firstmatrix M. The bandwidth of the second matrix N is proportional to√{square root over (S)}, so it is greatly reduced compared to thebandwidth of the first matrix M. FIG. 3 illustrates the second matrix Nobtained by applying this process to the first matrix M of FIG. 2. Mostof the non-zero elements are now close to the diagonal. The bandwidthhas been reduced from 460,320 (for the first matrix M of FIG. 2) to only1,922 (for the second matrix N of FIG. 3), which is a reduction by afactor of approximately 240. In certain embodiments, this reduction ofthe bandwidth advantageously speeds up the calculation of theeigenvalues of the matrix.

In principle, in certain embodiments, the modes of the fiber can befound by diagonalizing the second matrix N to find all of itseigenvalues. However, such an approach would require calculating andstoring a very large number of eigenvalues and eigenvectors, therebytaking a lot of time and memory. Furthermore, such an approach isgenerally wasteful because it calculates all of the eigenvalues, eventhose not of interest. More specifically, it would calculate all of theextremely high number of bulk (cladding) modes of a fiber, as well asits few core modes, whereas in general, knowledge of the cladding modesis not of interest. In certain other embodiments, Equation 7 is solvedby calculating only the few eigenvalues of interest. This restriction isnot unreasonable, since in general the primary interest is to find onlythe few core modes and a few bulk modes near the edges of the bandgapfor an air-core fiber. In certain such embodiments, the problem can thusbe restated as calculating the m eigenvalues of the second matrix N thatare closest to a given effective index value n₀, where m is a smallnumber. For example, to model the bandgap edges and effective indices ofthe modes within the bandgap of a particular air-core fiber at aparticular wavelength, the m₀=10 modes that are closest to n₀=1 can becalculated, since the core modes of an air-core fiber have effectiveindices close to 1.

Calculating the eigenvalues of a large sparse matrix that are closest toa given value is still a difficult task. To reduce the difficulty ofthis task, certain embodiments use the property that the m₀ eigenvaluesof a matrix N that are closest to some value n₀ are the same as thelargest (in amplitude) eigenvalues of a third matrix A, which can becalculated from the second matrix N using:A=(N−k ₀ ² n ₀ ² I)  (9)where I is the identity matrix (see, e.g., W. Zhi, R. Guobin, L. Shuqin,and J. Shuisheng, “Supercell lattice method for photonic crystalfibers,” Optics Express, 2003, Vol. 11, pages 980-991). This property isuseful in certain embodiments because the eigenvalues of maximumamplitude of a given matrix A can be calculated very fast by using theCourant-Fischer theorem (see, e.g., R. A. Horn and C. R. Johnson, MatrixAnalysis, Chapter 4, Cambridge University Press, London, 1983). Incertain embodiments, the Courant-Fischer theorem is used because itrestricts its eigenvalue search to small blocks of dimension m₀ withinthe matrix, which is much faster than identifying all the eigenvalues.

In certain embodiments, the method 100 further comprises shifting thesecond matrix N and inverting the shifted second matrix to form thethird matrix A in the operational block 140. Similar “shift-invert”techniques have previously been used in plane-wave expansions (see,e.g., W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Supercell latticemethod for photonic crystal fibers,” Optics Express, 2003, Vol. 11,pages 980-991), but have not been used as described herein. In certainsuch embodiments, the second matrix is shifted (e.g., by calculating anew matrix N−k₀ ²n₀ ²I), and is then inverted (e.g., by calculating theinverse of the matrix N−k₀ ²n₀ ²I) to calculate the third matrix A. Incertain embodiments, the inverse matrix is calculated using a standardLU decomposition (in which the matrix is expressed as the product of alower and an upper triangular matrices) to obtain the third matrix A(see, e.g., R. A. Horn and C. R. Johnson, “Matrix Analysis,” Chapter 3,Section 5, Cambridge University Press, 1990). This calculation is fastin certain embodiments because N−k₀ ²n₀ ²I is also a sparse matrix witha small bandwidth.

In certain embodiments, the method 100 further comprises calculating oneor more eigenvalues or eigenvectors of the third matrix A in theoperational block 150. The one or more eigenvalues or eigenvectorscorrespond to one or more modes of the waveguide. In certain embodimentsin which the waveguide comprises a PBF having a bandgap, thiscalculation comprises finding one or more modes of the PBF. The one ormore modes of the PBF comprises at least one core mode, surface mode,ring mode, or bulk mode near the bandgap. In certain embodiments, asstated above, the Courant-Fisher theorem is used to calculate only them₀ largest eigenvalues of A, which are the effective indices ofinterest. It can be seen that in certain embodiments described herein,the method 100 has transformed the problem from looking for a specificset of eigenvalues around a given value, which is a difficult problem,to looking for the largest eigenvalues of a transformed matrix, which isconsiderably faster.

In certain embodiments in which more detailed features in theelectromagnetic field distribution of a particular mode are desired(e.g., the tail in one of the air holes surrounding the core), certainembodiments utilize results obtained with a coarser grid as the newboundary conditions for extending the calculations to a finer grid, asis possible with other finite-difference or finite-element methods.Thus, this process can effectively zoom in to calculate theelectromagnetic fields in a specific portion of the waveguide. Incertain embodiments, this process can give access to very fine featuresof PBF modes.

FIG. 4 is another flow diagram illustrating an exemplary method 200 formodeling a PBF in accordance with certain embodiments described herein.The refractive index profile distribution of the fiber, including thestructure, the domain shape, and the boundary conditions, is provided asan input, as shown in FIG. 4 by the block 210. The wavelength ofinterest is also provided as an input, as shown in FIG. 4 by the block220. The refractive index profile distribution of the fiber is sampled,as shown in FIG. 4 by the arrow 230. The sparse first matrix M,representing the action of Maxwell's equations on the transversemagnetic field, is calculated, as shown in FIG. 4 by the block 240,using the sampled refractive index profile distribution as well as thewavelength and the boundary conditions.

As shown in FIG. 4 by the arrow 250, the first matrix M is rearranged toyield a sparse second matrix N, shown by the block 260, with a reducedbandwidth to speed up the extraction of the eigenvalues. The effectiveindex n₀ around which the PBF modes are to be calculated is provided asan input, as shown in FIG. 4 by the block 270. The second matrix N isshifted, shown in FIG. 4 by block 280, and the shifted matrix isinverted (e.g., using the LU decomposition shown in FIG. 4 by arrow 290)to provide the third matrix A, shown in FIG. 4 by block 300. The highesteigenvalues of the third matrix A are the eigenvalues closest to theeffective index n₀. The number of modes m₀ to be calculated is providedas an input, as shown in FIG. 4 by block 310, and the largesteigenvalues and eigenvectors of the third matrix A are calculated (e.g.,using the Courant-Fisher theorem), as shown in FIG. 4 by block 320. Thiscalculation provides the m₀ eigenvalues (modes) whose effective indicesare closest to the effective index n₀, as shown in FIG. 4 by the block330. In certain embodiments, this calculation also provides the m₀eigenvectors (all three components of the electric and magnetic fieldsof each mode) corresponding to each of these modes.

Certain embodiments described herein are useful in computer-implementedmodeling of the modal properties of waveguides (e.g., photonic-bandgapfibers). The general-purpose computers used for such modeling can take awide variety of forms, including network servers, workstations, personalcomputers, mainframe computers and the like. The code which configuresthe computer to perform such analyses is typically provided to the useron a computer-readable medium, such as a CD-ROM. The code may also bedownloaded by a user, from a network server which is part of alocal-area network (LAN) or a wide-area network (WAN), such as theInternet.

The general-purpose computer running the software will typically includeone or more input devices, such as a mouse, trackball, touchpad, and/orkeyboard, a display, and computer-readable memory media, such asrandom-access memory (RAM) integrated circuits and a hard-disk drive. Itwill be appreciated that one or more portions, or all of the code may beremote from the user and, for example, resident on a network resource,such as a LAN server, Internet server, network storage device, etc. Intypical embodiments, the software receives as an input a variety ofinformation concerning the waveguide (e.g., structural information,dimensions, refractive index profiles, wavelengths, number of modes,target index).

In certain embodiments, this mode-solving method 200 can be implementedstraightforwardly using any one of a number of commercial mathematicalsoftware programs. For example, as described more fully below, the modesolver method 200 can be implemented using MATLAB®, available from TheMathWorks of Natick, Mass., in which most of the desired mathematicaloperations are readily available. For example, MATLAB® provides variouslibraries of code (e.g., ARPACK and UMFPACK) that compute the largesteigenvalues of a matrix using the Courant-Fisher theorem mentionedabove. Using such built-in features of commercially-availablemathematical software programs and as a result of the relativesimplicity of certain embodiments of the mode-solving method describedherein, certain embodiments described herein utilize only minimalprogramming, with the section of code corresponding to the mode solvingcontaining less than approximately 200 lines of code.

Certain embodiments described herein advantageously provide considerablyspeedier and easier simulations of PBFs than do other waveguidesimulators. In addition, certain embodiments described hereinadvantageously do not require a supercomputer. For example, in certainembodiments, the mode solving method is performed on a personal computerwith a 3.2-GHz Pentium® IV processor and 4 gigabytes of RAM. In certainsuch embodiments, the mode solver finds the propagation constants andfields of 20 modes in a given fiber in approximately four minutes perwavelength for a rectangular grid with 500 points on each side.Conversely, applying the MPB code (cited above), which is based on aplane-wave expansion technique, to the same problem takes approximatelyone hour using sixteen 2-GHz parallel processors, and such a calculationsimply cannot be done in a reasonable time on a personal computer.

In certain embodiments, the simulator code which implements the modesolving method is very short (e.g., less than 200 lines). In certainembodiments, the method advantageously samples the exact refractiveindex distribution. In certain embodiments, the method uses thewavelength as the variable parameter, which is a more intuitive approachtraditionally used to model conventional fibers and optical waveguides.In certain embodiments, the method does not calculate all the modes, butonly calculates desired modes, and can be used with a variety ofboundary conditions. In certain embodiments, the method modelsrefractive index distributions with negative or complex values, whichmakes it possible to estimate mode propagation losses using perfectlymatching boundary conditions.

In the following description, the performance of an exemplary embodimentis illustrated by applying it to model the propagation constants (orequivalently, the effective mode indices) and intensity profiles (orequivalently the electric and magnetic field spatial distributions) ofselected modes of an air-core PBF with circular air holes in atriangular lattice in silica. The fiber cross-section used in thesimulation is shown in FIG. 5, which has a circular air-hole with aradius ρ=0.47Λ and a radius of the circular air core R=0.8Λ. The blackareas of FIG. 5 represent silica with regions of air therebetween. ThePBF structure of FIG. 5 exhibits a bandgap for ρ>0.43Λ. In the followingdescription, the modeling was done using periodic boundary conditions.In particular, the simulation domain (supercell) represented in FIG. 5was repeated periodically to tile the entire space, and the modes ofthis periodic waveguide were calculated. The separation distance betweentwo fiber cores in this periodic tiling was selected to be large enough(e.g., 9Λ) to ensure negligible interaction between neighboring fibercores and thus to ensure accurate predictions. The simulation domain wastherefore a rectangle 9Λ wide (slightly under five rings), a sizecommonly used in simulations because it offers a good compromise betweenaccuracy on one hand and computing time and memory requirements on theother hand. The sampling of the fiber refractive index distribution, andthus the digitization of the fiber features (e.g., circular air core andcladding air holes) was done with a spatial resolution of Λ/50. Thisfinite resolution accounts for the small irregularities visible at theedge of the holes in FIG. 5.

FIG. 6 shows the dispersion curve of a first PBF with a core radiusR=1.0Λ calculated using an exemplary method compatible with certainembodiments described herein. The bandgap extends approximately fromλ=0.56Λ to λ=0.64Λ. This bandgap supports two fundamental core modes(Gaussian-like HE₁₁ modes polarized along the fiber axes) and does notsupport surface modes, as expected from the particular choice of coreprofile and radius (see, e.g., M. J. F. Digonnet, H. K. Kim, J. Shin, S.H. Fan, and G. S. Kino, “Simple geometric criterion to predict theexistence of surface modes in air-core photonic-bandgap fibers,” OpticsExpress, Vol. 12, No. 9, 3 May 2004, pages 1864-1872; and H. K. Kim, J.Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-corephotonic-bandgap fibers free of surface modes,” IEEE Journal of QuantumElectronics, Vol. 40, No. 5, May 2004, pages 551-556). As is well-known,the fundamental core modes are mainly concentrated in air. FIG. 7 showsa contour map of the intensity profile I=|E_(x)|²+|E_(y)|² of one of thefundamental core modes for a core radius of 0.8Λ. The rightmost of thetwo views of FIG. 7 is a magnified version of the leftmost of the twoviews. FIG. 8 shows an intensity profile (cut along x=0) of thefundamental core mode of FIG. 7. The fundamental core mode is notperfectly six-fold symmetric as expected from the photonic crystalsymmetry due to its degeneracy with a mode which is polarizedorthogonally. However, the fundamental core mode is symmetric about boththe vertical axis (x=0) and the horizontal axis (y=0). Note that thisprofile exhibits peaks at the air-silica interface at the core boundary(y=±1.5Λ, as shown in FIG. 8). Such features are examples of the finemodal features that can be resolved by certain embodiments describedherein which sample the exact refractive index distribution.

The convergence of this exemplary method is illustrated by FIG. 9, whichshows the convergence of the fundamental core mode effective index as afunction of the number of grid points for the PBF structure of FIG. 5(shown in FIG. 9 by the diamond-shaped data points, and referring to theleftmost vertical axis). Convergence to the fifth decimal place is shownin FIG. 9 for a step size of Λ/50 (approximately 450 points per side)for the PBF structure of FIG. 5.

To assess its accuracy, the exemplary method was used to calculate theeffective index of the fundamental mode of the air-assisted fiberpreviously calculated using the multipole method, as described by Z. Zhuand T. G. Brown, in “Full-vectorial finite-difference analysis ofmicrostructured optical fibers,” Optics Express, 2002, Vol. 10, pages853-864 (“Zhu et al.”) and illustrated by FIG. 3 of Zhu et al. Theresults of the calculation using the exemplary method (shown in FIG. 9by the square-shaped data points, and referring to the rightmostvertical axis) were compared to the values of the effective indexcalculated using the multipole method as reported by Zhu et al. (shownin FIG. 9 by the dashed line, and referring to the rightmost verticalaxis). Table 1 lists the values obtained using the exemplary method withtwo different grid sizes, and the values previously reported for themultipole method by Zhu et al.

TABLE 1 Exemplary Exemplary method; method; Multipole Solution 160points 256 points method of Method per side per side Zhu et al.Fundamental 1.43536053 1.43535983 1.4353607 mode effective indexAs shown in Table 1, accuracy to the fifth decimal place is achieved bythe exemplary method for a grid of 160 points per side. The modeeffective index calculated with the exemplary method using 256 gridpoints (1.43535983) differs from the value calculated with the multipolemethod (1.4353607) by only about 9×10⁻⁷. FIG. 9 also illustrates theconvergence for the exemplary method for increasing grid sizes for thefiber illustrated by FIG. 3 of Zhu et al. (shown in FIG. 9 bysquare-shaped data points). The convergence achieved is better thanseventh decimal (for 550 grid points per side) and the accuracy is ofcomparable order. These results confirm both the accuracy and theconvergence of the exemplary method.

FIG. 10 shows the calculated dispersion diagram of a second PBF having acore radius of R=1.15Λ. This fiber exhibits surface modes, which isconsistent with previously published results (see, e.g., M. J. F.Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S. Kino, “Simplegeometric criterion to predict the existence of surface modes inair-core photonic-bandgap fibers,” Optics Express, Vol. 12, No. 9, 3 May2004, pages 1864-1872; and H. K. Kim, J. Shin, S. H. Fan, M. J. F.Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap fibersfree of surface modes,” IEEE Journal of Quantum Electronics, Vol. 40,No. 5, May 2004, pages 551-556). As shown in the intensity contour mapof the exemplary surface mode of FIG. 11, these surface modes arelocalized at the interface between the air core and first layer ofcladding air holes. The fields of these surface modes have their maximain the silica and are evanescent in air. The surface mode shown in FIG.11 exhibits the six-fold symmetry expected from the fiber symmetry.These results are consistent with previously reported PBF mode behavior.

For simulations of PBFs, certain embodiments advantageously minimize thetotal computation time by solving for a large number of modes of thestructure (e.g., approximately 40) around a target effective index(e.g., n₀=1 or 0.99) until a core mode is found. Since the effectiveindices of core modes are not known prior to the simulation, in certainembodiments, this calculation of a large number of modes is performed tofind the core modes at a particular wavelength. Once the core modeeffective indices have been determined at a given wavelength, each ofthese effective indices can be used as the target effective index of thecorresponding mode at the next (or nearby) wavelength, or anextrapolation (e.g., linear) can be used to estimate the targeteffective index. When the effective index of a given mode has beencalculated at two (or more) wavelengths, either a linear or a splineinterpolation technique (or an equivalent technique) can be used toestimate the target effective index. In certain embodiments, the numberof modes solved at this wavelength can then be lowered (e.g., to lessthan 20, since in general only the core modes, surface modes, and a fewbulk modes at the edges of the bandgap are of interest). Certain suchembodiments advantageously reduce the computation time by tracking aparticular mode of interest, and advantageously center the calculatedmodes around the core modes.

Certain embodiments described herein provide a new photonic-bandgapfiber mode solver which simulates the modes of air-core photonic-bandgapfibers of arbitrary index profiles quickly using a personal computerwhile reducing the amount of data storage used by about two orders ofmagnitude. In certain embodiments, the effective indices and fieldprofiles of the modes of a typical PBF at a particular wavelength can becalculated in approximately 10 seconds per mode on a personal computerwith 3.2 gigabytes of RAM. In certain embodiments, these substantialimprovements are accomplished by (1) solving a vectorialtransverse-magnetic-field equation in a matrix form, which can be donequickly using a sparse matrix; (2) re-arranging the matrix elements toreduce the matrix bandwidth, which further speeds up the calculation byanother factor of at least one order of magnitude; and (3) solvingexclusively for the modes of interest. Certain embodiments describedherein are advantageously simple to program (e.g., about 200 lines ofMATLAB® code), can be advantageously used in conjunction with one ofseveral types of boundary conditions, and are advantageously applicableto calculate the modes of any optical fiber of waveguide, regardless ofthe complexity of its index profile.

Various embodiments have been described above. Although this inventionhas been described with reference to these specific embodiments, thedescriptions are intended to be illustrative of the invention and arenot intended to be limiting. Various modifications and applications mayoccur to those skilled in the art without departing from the true spiritand scope of the invention as defined in the appended claims.

Listed below are exemplary MATLAB® code portions which can be used toimplement a method compatible with certain embodiments described hereinfor calculating the modal properties of a waveguide. In particular,exemplary code portion “mode_hex” is compatible with periodic boundaryconditions, and exemplary code portion “mode_rec” is compatible withrectangular boundary conditions.

The exemplary code portions listed herein contain material which issubject to copyright protection. The copyright owner has no objection tothe facsimile reproduction by anyone of the patent document or thepatent disclosure, as it appears in the Patent and Trademark Office fileor records, but otherwise reserves all copyright rights whatsoever.

Exemplary Code Portion “mode_hex”

Exemplary code portion “mode_hex” can be used to solve for eigenvaluesand eigenvectors for hexagonal boundaries:

function nakapuf=mode_hex(filen,wa1,wa2,was,modes,nint) %2D cartesiancoordinates mode solver %Hexagonal boundary %Sparse matriximplementation %Transverse magnetic field is solved using its eigenvalueequation %clear %filen=input(‘Refractive index filename: ’,‘s’) ;load(filen) % wa1=input (‘Start wavelength: ’) ; % % wa2=input (‘Finishwavelength: ’) ; % % was=input(‘Wavelength step: ’) ; %modes=input(‘Number of modes to calculate: ’) ; % %nint=input(‘Effective index of interest: ’) ; lam=wa1:was:wa2 ;outb=strcat(filen,‘_buffer.mat’) ; S=length(K) ; clear si[Lx,Ly]=grad_h(log(refh.{circumflex over( )}2),Xh,Yh,Pxp,Pxm,Pyp,Pym,sx,sy) ; ‘done phase 1’ form0=1:length(lam), load(filen) lamb=lam(m0) ; k0=2*pi/lamb ;subname=strcat(filen,‘_mode_’,num2str(m0)) ; l=1 ; ind=1:S ; ind0=0*ind; i=[ind ind+S ind ind ind ind ind ind ind+S ind+S ind+S ind+S ind+Sind+S] ; j=[ind ind+S Pxp′ Pxm′ Pyp′ Pym′ Pxp′+S Pxm′+S Pxm′+S Pxp′+SPyp′+S Pym′+S Pyp′ Pym′] ; s=[−2/sx{circumflex over( )}2−2/sy{circumflex over ( )}2+k0{circumflex over( )}2*refh.{circumflex over ( )}2 −2/sx{circumflex over( )}2−2/sy{circumflex over ( )}2+k0{circumflex over ( )}2*refh.{circumflex over ( )}2 1/sx{circumflex over ( )}2+ind0 1/sx{circumflexover ( )}2+ind0 1/sy{circumflex over( )}2−0.5*Ly(ind)./sy 1/sy{circumflex over( )}2+0.5*Ly(ind)./sy 0.5*Ly(ind)./sx − 0.5*Ly(ind)./sx 1/sx{circumflexover ( )}2+0.5*Lx(ind)./sx 1/sx{circumflex over( )}2−0.5*Lx(ind)./sx 1/sy{circumflex over ( )}2+ind0 1/sy{circumflexover ( )}2+ind0 0.5*Lx(ind)./sy −0.5*Lx(ind)./sy] ; clear Pxp Pxm PypPym ind0 ind ‘done phase 2’ M0=sparse(i,j,s) ; ‘done phase 3’p=symrcm(M0) ; ‘done phase 4’ M1=M0(p,p) ; save(outb,‘M0’,‘i’,‘j’,‘s’)clear M0 i j s ‘done phase 5’ [V,D]=eigs(M1,modes,nint{circumflex over( )}2*k0{circumflex over ( )}2) ; ‘done phase 6’ d=diag(D) ;[d0,li]=sort(d) ; d=flipdim(d0,l) ; for i0=l:length(d), nef =sqrt(d(i0)/k0{circumflex over ( )}2) ; NEFF(m0,i0) = nef ; neff(i0)=nef; W(p,i0)=V(:,li(length(d)+1−i0)) ; beta(m0,i0) = k0*nef ; end clear Vsave(subname,‘W’,‘neff’,‘lamb’,‘ref’,‘xv’,‘yv’) ; clear neff W ‘donephase 7’ m0 end outp=strcat(filen,‘_solut.mat’) ; save(outp,‘NEFF’,‘beta’,‘lam’,‘ref’) returnExemplary Code Portion: “mode_rec”

Exemplary code portion “mode_rec” can be used to solve for eigenvaluesand eigenvectors for rectangular boundaries:

function nakapuf=mode_rec(filen,wa1,wa2,was,modes,nint) %2D cartesiancoordinates mode solver %Sparse matrix implementation %Transversemagnetic field is solved using its eigenvalue equation % clear % %filen=input(‘Refractive index filename: ’,‘s’) ; load(filen) %wa1=input(‘Start wavelength: ’) ; % % wa2=input(‘Finish wavelength: ’) ;% % was=input(‘Wavelength step: ’) ; % % modes=input(‘Number of modes tocalculate: ’) ; % % nint=input(‘Effective index of interest: ’) ;lam=wa1:was:wa2 ; %contains 2D refractive index matrix, spatial stepsize s0 si=size(ref) ; Nx=si(1) ; Ny=si(2) ; clear si[Lx,Ly]=gradxy(log(ref.{circumflex over ( )}2),sx,sy) ; ‘done phase 1’%generation of sparse matrix i,j,s for m0=l:length(lam) , lamb=lam(m0) ;k0=2*pi/lamb ; subname=strcat(filen,‘_mode_’,num2str(m0)) ; l=1 ;S=Nx*Ny ; ind=1:S ; [Y,X]=meshgrid(1:Ny,1:Nx) ; Z=X+Nx*(Y−1) ;Zin=Z((2:Nx−1),(2:Ny−1)) ; indi=Zin(1:(Nx−2)*(Ny−2)) ; indi0 = 0*indi ;ib=[ind ind+S indi indi indi indi indi indi indi+S indi+S indi+S indi+Sindi+S indi+S] ; jb=[ind ind+S indi+1 indi−1 indi+Nx indi−Nx indi+1+Sindi−1+S indi−1+S indi+1+S indi+Nx+S indi−Nx+S indi+Nx indi−Nx] ;sb=[−2/sx{circumflex over ( )}2−2/sy{circumflex over ( )}2+k0{circumflexover ( )}2*ref(ind).{circumflex over ( )}2 −2/sx{circumflex over( )}2−2/sy{circumflex over ( )}2+k0{circumflex over( )}2*ref(ind).{circumflex over ( )}2 1/sx{circumflex over( )}2+indi0 1/sx{circumflex over ( )}2+indi0 1/sy{circumflex over( )}2−0.5*Ly(indi)./sy 1/sy{circumflex over ( )}2+0.5*Ly(indi)./sy0.5*Ly(indi)./sx −0.5*Ly(indi)./sx 1/sx{circumflex over( )}2+0.5*Lx(indi)./sx 1/sx{circumflex over ( )}2−0.5*Lx(indi)./sxupb=1+Nx*(1:Ny−2) ; downb=Nx+Nx*(1:Ny−2) ; leftb=(2:Nx−1) ;rightb=Nx*(Ny−1) + (2:Nx−1) ; b0=0*upb ; b0y=0*leftb ; ‘done phase 2’i=[ib upb upb upb upb upb+S upb+S upb+S upb+S upb+S] ; j=[jb upb+1upb+Nx upb−Nx upb+1+S upb+1+S upb+Nx+S upb−Nx+S upb+Nx upb−Nx] ; s=[sb1/sx{circumflex over ( )}2+b0 1/sy{circumflex over ( )}2−0.5*Ly(upb)./sy1/sy{circumflex over ( )}2+0.5*Ly(upb)./sy 0.5*Ly(upb)./sx1/sx{circumflex over ( )}2−0.5*Lx(upb)./sx 1/sy{circumflex over ( )}2+b01/sy{circumflex over ( )}2+b0 0.5*Lx(upb)./sy −0.5*Lx(upb)./sy] ; ib1=i; jb1=j ; sb1=s ; i=[ib1 downb downb downb downb downb+S downb+S downb+Sdownb+S downb+S] ; j=[jb1 downb−1 downb+Nx downb−Nx downb−1+S downb−1+Sdownb+Nx+S downb−Nx+S downb+Nx downb−Nx] ; s=[sb1 1/sx{circumflex over( )}2+b0 1/sy{circumflex over ( )}2−0.5*Ly(downb)./sy 1/sy{circumflexover ( )}2+0.5*Ly(downb)./sy  − 0.5*Ly(downb)./sx 1/sx{circumflex over( )}2+0.5*Lx(downb)./sx 1/sy{circumflex over ( )}2+b0 1/sy{circumflexover ( )}2+b0 0.5*Lx(downb)./sy − 0.5*Lx(downb)./sy] ; ib2=i ; jb2=j ;sb2=s ; i=[ib2 leftb leftb leftb leftb leftb leftb+S leftb+S leftb+Sleftb+S] ; j=[jb2 leftb+1 leftb−1 leftb+Nx leftb+1+S leftb−1+S leftb−1+Sleftb+1+S leftb+Nx+S leftb+Nx] ; s=[sb2, 1/sx{circumflex over ( )}2+b0y,1/sx{circumflex over ( )}2+b0y, 1/sy{circumflex over( )}2−0.5*Ly(leftb)./sy, 0.5*Ly(leftb)./sx, − 0.5*Ly(leftb)./sx,1/sx{circumflex over ( )}2+0.5*Lx(leftb)./sx, 1/sx{circumflex over( )}2−0.5*Lx(leftb)./sx, 1/sy{circumflex over ( )}2+b0y,0.5*Lx(leftb)./sy] ; ib3=i ; jb3=j ; sb3=s ; i=[ib3 rightb rightb rightbrightb rightb rightb+S rightb+S rightb+S rightb+S] ; j=[jb3 rightb+1rightb−1 rightb-Nx rightb+1+S rightb−1+S rightb−1+S rightb+1+Srightb−Nx+S rightb−Nx] ; s=[sb3 1/sx{circumflex over ( )}2+b0y1/sx{circumflex over ( )}2+b0y 1/sy{circumflex over( )}2+0.5*Ly(rightb)./sy 0.5*Ly(rightb)./sx − 0.5*Ly(rightb)./sx1/sx{circumflex over ( )}2+0.5*Lx(rightb)./sx 1/sx{circumflex over( )}2−0.5*Lx(rightb)./sx 1/sy{circumflex over ( )}2+b0y −0.5*Lx(rightb)./sy] ; ib4=i ; jb4=j ; sb4=s ; indur=1−Nx+S ; indll=Nx ;indlr=S ; i=[ib4 1 1 1 1+S 1+S 1+S indur indur indur indur+S indur+Sindur+S indll indll indll indll+S indll+S indll+S indlr indlr indlrindlr+S indlr+S indlr+S] ; j=[jb4 2 1+Nx 1+Nx+S 2+S 1+Nx+S 1+Nx indur+1indur−Nx indur+1+S indur+1+S indur−Nx+S indur−Nx indll−1 indll+Nxindll−1+S indll−1+S indll+Nx+S indll+Nx indlr− 1 indlr−Nx indlr−1+Sindlr−1+S indlr−Nx+S indlr−Nx] ; s=[sb4 1/sx{circumflex over ( )}2 1/sy{circumflex over ( )}2−0.5*Ly(1)/sy 0.5*Ly(1)/sy 1/sx{circumflexover ( )}2−0.5*Lx(1)/sx  1/sy{circumflex over ( )}2 0.5*Lx(1)/sy1/sx{circumflex over ( )}2  1/sy{circumflex over( )}2+0.5*Ly(indur)/sy 0.5*Ly(indur)/sx 1/sx{circumflex over ( )}2−0.5*Lx(indur)/sx 1/sy{circumflex over ( )}2 −0.5*Lx(indur)/sy 1/sx{circumflex over ( )}2  1/sy{circumflex over( )}2−0.5*Ly(indll)/sy − 0.5*Ly(indll)/sx 1/sx{circumflex over( )}2+0.5*Lx(indll)/sx  1/sy{circumflex over( )}2 0.5*Lx(indll)/sy 1/sx{circumflex over ( )}2 1/sy{circumflex over( )}2+0.5*Ly(indlr)/sy  −0.5*Ly(indlr)/sx 1/sx{circumflex over( )}2+0.5*Lx(indlr)/sx 1/sy{circumflex over ( )}2  − 0.5*Lx(indlr)/sy] ;clear ib jb sb ib1 jb1 sb1 ib2 ib3 ib4 jb2 jb3 jb4 sb2 sb3 sb4 ‘donephase 3’ %1 M0=sparse(i,j,s) ; ‘done phase 4’ p=symrcm(M0) ; ‘done phase5’ M1=M0(p,p) ; %spy(M1) ‘done phase 6’ outb=strcat(filen,‘_buffer.mat’); save(outb,‘M0’,‘i’,‘j’,‘s’) clear M0 i j s[V,D]=eigs(M1,modes,nint{circumflex over ( )}2*k0{circumflex over ( )}2); ‘done phase 7’ d=diag(D) ; [d0,li]=sort(d) ; d=flipdim(d0,1) ; fori0=l:length(d), nef = sqrt(d(i0)/k0{circumflex over ( )}2) ; NEFF(m0,i0)= nef ; neff(i0)=nef ; W(p,i0)=V(:,li(length(d)+1−i0)) ;%[hx,hy]=remap(W(:,i0)) ; beta(m0,i0) = k0*nef ; endsave(subname,‘W’,‘neff’,‘lamb’,‘ref’,‘xv’,‘yv’) ; endoutp=strcat(filen,‘_solut.mat’) ; save(outp,‘NEFF’, ‘beta’,‘lam’,‘ref’)

1. A method for modeling one or more electromagnetic field modes of awaveguide, the method comprising: sampling a two-dimensionalcross-section of the waveguide; and using a computer to perform thesteps comprising: calculating a first matrix using the sampledtwo-dimensional cross-section of the waveguide, the first matrixcomprising a plurality of elements and having a first bandwidth, theplurality of elements of the first matrix representing an action ofMaxwell's equations on a transverse magnetic field within the waveguide;rearranging the plurality of elements of the first matrix to form asecond matrix having a second bandwidth smaller than the firstbandwidth; shifting the second matrix and inverting the shifted secondmatrix to form a third matrix; calculating one or more eigenvalues oreigenvectors of the third matrix corresponding to one or more modes ofthe waveguide; and calculating one or more mode propagation lossescorresponding to the one or more modes of the waveguide.
 2. The methodof claim 1, wherein the waveguide comprises a photonic-bandgap fiber. 3.The method of claim 1, wherein the waveguide comprises an air-corephotonic-bandgap fiber.
 4. The method of claim 1, wherein the waveguidehas a refractive index profile which is translation invariant along alongitudinal axis of the waveguide.
 5. The method of claim 1, whereinsampling the two-dimensional cross-section of the waveguide comprisesdigitizing a refractive index profile of the waveguide.
 6. The method ofclaim 1, wherein the waveguide comprises an air-core photonic-bandgapfiber having a longitudinal axis, an air core and a cladding structurecomprising air holes and intervening membranes, and sampling thetwo-dimensional cross-section of the waveguide comprises digitizing theair core and cladding structure in a planar cross-section which isperpendicular to the longitudinal axis.
 7. The method of claim 6,wherein sampling the two-dimensional cross-section of the waveguide isperformed over an area corresponding to a minimum cell that is afundamental component of the cladding structure.
 8. The method of claim1, wherein calculating the first matrix comprises defining boundaryconditions.
 9. The method of claim 1, wherein calculating the firstmatrix comprises discretizing an eigenvalue equation satisfied by thetransverse magnetic field, the discretizing being performed usingMaxwell's equations as expressed by${{\overset{->}{\nabla}{\times \overset{->}{E}}} = \frac{- {\partial\overset{->}{B}}}{\partial t}},{{\overset{->}{\nabla}{\times \overset{->}{H}}} = \frac{\partial\overset{->}{D}}{\partial t}},{{{and}\mspace{14mu}{\overset{->}{\nabla}{\cdot \overset{->}{H}}}} = 0.}$10. The method of claim 9, wherein discretizing the eigenvalue equationfurther comprises index-averaging over each discretization pixelstraddling an air-core boundary.
 11. The method of claim 9, wherein thewaveguide comprises an air-core photonic-bandgap fiber having alongitudinal axis, an air core and a cladding structure comprising airholes and intervening membranes, and wherein discretizing the eigenvalueequation comprises sampling two components of the transverse magneticfield in a planar cross-section of the waveguide.
 12. The method ofclaim 11, wherein sampling the two components of the transverse magneticfield is performed over an area corresponding to a minimum cell that isa fundamental component of the cladding structure.
 13. The method ofclaim 1, wherein calculating the first matrix comprises sampling alinear operator corresponding to the action of Maxwell's equations on atransverse magnetic field within the waveguide.
 14. The method of claim1, wherein inverting the shifted second matrix comprises using an LUdecomposition.
 15. The method of claim 1, wherein calculating one ormore eigenvalues or eigenvectors of the third matrix comprisescalculating only a selected number of the largest eigenvalues of thethird matrix.
 16. The method of claim 1, wherein calculating one or moreeigenvalues or eigenvectors of the third matrix comprises usingfinite-difference or finite-element calculations.
 17. The method ofclaim 1, wherein the waveguide comprises a photonic-bandgap fiber havinga bandgap and calculating one or more eigenvalues or eigenvectors of thethird matrix comprises finding one or more modes of the photonic-bandgapfiber, the one or more modes comprising at least one core mode, surfacemode, ring mode, or bulk mode near the bandgap.
 18. A computer-readablemedium having instructions stored thereon which cause a general-purposecomputer to perform the method of claim
 1. 19. A computer system formodeling one or more electromagnetic field modes of a waveguide, thecomputer system comprising: a processor comprising: means for sampling atwo-dimensional cross-section of the waveguide; means for calculating afirst matrix having a first bandwidth using the sampled two-dimensionalcross-section of the waveguide, the first matrix comprising a pluralityof elements representing an action of Maxwell's equations on atransverse magnetic field within the waveguide; means for rearrangingthe plurality of elements of the first matrix to form a second matrixhaving a second bandwidth smaller than the first bandwidth; means forshifting the second matrix and inverting the shifted second matrix toform a third matrix; means for calculating one or more eigenvalues oreigenvectors of the third matrix corresponding to one or more modes ofthe waveguide; and means for calculating one or more mode propagationlosses corresponding to the one or more modes of the waveguide.